this may seem like a simple question but how does one know that separation of variables for solving linear PDE's will work. What i mean is that it seems to pick out a form of the solution to a given problem (I have heard that linear PDE's have an infinite number of functions of a particular form, e.g. for the wave equation the solution is of the form f(x-vt) + g(x+vt)). I can understand that for problems in e&m the separation of variables technique picks out a particular form (like for a cartesian coordinates for laplace's equation for a box, the solutions come out to be sines and cosines), but what about linear PDE's in finance (like the Black scholes equation). Thanks in advance to anyone who can clarify this.